Method for locating of single-phase-to-ground faults of ungrounded power distribution systems

ABSTRACT

The method determines shunt caused residual voltages and fault caused residual voltages on the upstream bus and the downstream bus of the line segment within a faulty feeder section of a faulty feeder. The line segment is designated as a faulty line segment when a reference angle of a faulty phase is between a first angle of a difference between an angle of the fault caused residual voltage and an angle of the shunt caused residual voltage on the upstream bus and a second angle of a difference between an angle of the fault caused residual voltage and an angle of the shunt caused residual voltage on the downstream bus. The location of the fault is determined at a point on the faulty line segment with a difference between the angles of the fault and the shunt caused residual voltages in phase with the reference angle.

FIELD OF THE INVENTION

The present invention relates generally to power distribution systems,and more particularly to detecting and locating of a fault in theungrounded power distribution systems.

BACKGROUND OF THE INVENTION

Ungrounded power distribution systems are widely used, especially atmedium voltage levels, e.g., less than 50 kV. Compared with the groundeddistribution systems, the ungrounded systems do not have neural wires toconnect with the ground, and they are connected to ground throughphase-to-ground capacitances of power lines. When asingle-phase-to-ground fault occurs, the fault currents of ungroundeddistribution systems are less than normal load currents, thus the systemcan continue to operate until the fault is corrected.

However, as a result of the fault, the power lines of the distributionsystems experience over-voltages, which can damage the lines when thefault is not corrected in a timely manner. Thus, detecting and locatingof a fault is important for the safe and stable operation of ungroundeddistribution systems.

Several methods have been used for locating single-phase-to-groundfaults in ungrounded distribution systems. For example, a methoddescribed in U.S. Pat. No. 6,721,671 for determining a section of thesystem having a fault uses a directional element to determine faults onungrounded power systems, which following enablement under selectedinput current conditions, determined zero sequence impedance, inresponse to values of zero sequence voltage and zero sequence current.

Another method, described in US 2003/0085715, introduces a measurementsignal having a measurement frequency on the line having a fault. Thefault location is determined for a selected segment based on a measuredresidual current corresponding to the measurement signal, and apredetermined relative impedance of the power distribution, system.However, usage of additional frequency measurements is not optimal forsome applications.

Accordingly, there is a need for determining locations ofsingle-phase-to-ground faults in the ungrounded power distributionsystems.

SUMMARY OF THE INVENTION

Various embodiments of invention determine the location of a boltedsingle-phase-to-ground fault in an ungrounded power distribution systembased on measurements collected during the fault. For example, themeasurements can be determined by the measuring units or sensorsinstalled at the feeder breakers and switches.

Some embodiments are based on a realization that a possible faulty areacan be narrowed down into a small section of a feeder if multiplemeasuring devices are used for feeders in a substation. Some embodimentsof the invention uses the measurements collected from the feederbreakers at the roots of feeders, and switches with sensors along thefeeders. In some embodiments of the invention, the faulty phase, faultyfeeder and faulty feeder section are first determined based on thevoltage and current measurements during, the fault, and then thepossible faulty line segment and faulty location are determined bydetailed analysis of voltage and current distribution in the faultyfeeder section.

A faulty line segment can be determined by comparing angle of thephase-to-ground voltage of the faulty phase against a reference angle ofthe faulty phase if assuming that the fault current flows across theline segment instead of entering into ground through a location withinthe line segment. If the value of the reference angle of the faultyphase is between the angles of the voltages on the bus terminals formingthe line segment, i.e., an upstream bus and a downstream bus, then thisline segment is faulty. However, in order to determine the voltage onthe buses, the loads of the buses should be determined, which can be adifficult task. Thus, it is desired to avoid the determination of theloads.

Some embodiments are based on a realization that a difference betweenthe fault caused residual voltage and the shunt caused residual voltageof the bus can approximate the voltage on the faulty phase of the bus.The fault caused residual voltage is a sum of the voltages of all threephases at the time of the fault determined under assumption that thecurrent flows across the line segment. This assumption leads to a changeof sign of the difference between the angle of the voltage on thedownstream bus and a reference angle of the faulty phase comparing to avoltage on the upstream bus of the faulty line.

A shunt caused residual voltage is a sum of the voltages of all threephases without fault determined based on shunt current, i.e., withoutthe fault current. Thus, the difference between the fault causedresidual voltage and the shunt caused residual can approximate thevoltage of the faulty phase, but because those voltages includes sum ofthe voltages of the phases, and the loads are DELTA-connected, the loadsof different phases cancel each other and thus do not have to bedetermined. Accordingly, determination of the faulty line is simplified.

Moreover, the difference between the fault caused residual voltage andthe shunt caused residual voltage at the location of the fault is inphase with a reference angle of the faulty phase. Thus, various locationof the faulty line segment can be tested with this equality to determinethe location of the fault.

Some embodiments are based on another realization that the variations ofresidual voltages upstream and downstream to the faulty location showsdifferent patterns due to the direction change of residual currentsaround the fault location, so some embodiments of the inventiondetermine the fault location based on the variations of residualvoltages along the line segments. Some embodiments of the inventiondefines two types of residual voltages, one is called shunt causedresidual voltages which is used to describe the residual voltagedistribution of the feeder section under an un-faulty condition withgiven measured voltages at the boundaries of the feeder section, and theother is called the fault caused residual voltages which are used todescribe the residual voltage distribution under a faulty condition thatoccurred downstream to the location of interest. The faulty line segmentis identified when the phase angles of the difference between tworesidual voltages determined at two terminal buses of a line segment areat different sides of the faulty phase based reference axis. The faultlocation is determined by finding a location along the faulty line thatthe phase angle of difference of two residual voltages are in phase withthe reference angle.

Yet another realization that sonic embodiments of the invention arebased upon is that the residual currents and residual voltages aredominantly caused by the shunt admittances of line segments for anungrounded distribution system when the asymmetry of distribution linesis ignored, thus some embodiments of the invention determines the faultlocation solely based on the voltage and current measurements during thefault, and series impedance and shunt admittances of line segments inthe feeder sections. Some embodiments of the invention do not use anyinformation or measurements regarding the load demands of the system, orthe pre-fault conditions in the system. By doing so, the efforts formeasurement collecting and processing are significantly reduced.

Accordingly, one embodiment discloses a method for determining alocation of a fault in an ungrounded power distribution system, whereinthe power distribution system includes a set of feeders connected to asubstation, wherein each feeder includes a set of loads connected byline segments and each line segment includes an upstream bus and adownstream bus, and the fault is a bolted single-phase-to-ground fault.The method included determining shunt caused residual voltages on theupstream bus and the downstream bus of the line segment within a faultyfeeder section of a faulty feeder; determining fault caused residualvoltages on the upstream bus and the downstream bus of the line segment;designating the line segment as a faulty line, segment when a referenceangle of a faulty phase is between a first angle of a difference betweenan angle of the fault caused residual voltage and an angle of the shuntcaused residual voltage on the upstream bus and a second angle of adifference between an angle of the fault caused residual voltage and anangle of the shunt caused residual voltage on the downstream bus; anddetermining a location of a point on the faulty line segment with adifference between the angle of the fault caused residual voltage andthe angle of the shunt caused residual voltage in phase with thereference angle of the faulty phase as the location of the fault. Thesteps of the method can be performed by a processor.

Another embodiment discloses a system for determining a location of afault in an ungrounded power distribution system, wherein the powerdistribution system includes a set of feeders connected to a substation,wherein each feeder includes a set of loads connected by line segmentsand each line segment includes an upstream bus and a downstream bus, andthe fault is a bolted single-phase-to-ground fault. The system includesa processor for determining, shunt caused residual voltages on theupstream bus and the downstream bus of the line segment within a faultyfeeder section of a faulty feeder; determining fault caused residualvoltages on the upstream bus and the downstream bus of the line segment;determining a faulty line segment as a line segment with a value of areference angle of a faulty phase between a first angle and a secondangle, wherein the first angle equals a difference between an angle ofthe fault caused residual voltage and an angle of the shunt causedresidual voltage on the upstream bus, and Wherein the second angleequals a difference between an angle of the fault caused residualvoltage and an angle of the shunt caused residual voltage on thedownstream bus; and determining a location of a point on the faulty linesegment with a difference between the angle of the fault caused residualvoltage and the angle of shunt caused residual voltage in phase with thereference angle of the faulty phase as the location of the fault.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of an exemplar ungrounded distribution system;

FIG. 2 is a block diagram of a method for locating thesingle-phase-to-ground faults of ungrounded systems according to someembodiments of invention;

FIG. 3 is a schematic ala feeder section, with a breaker or switch withsensor as an importing measuring device, and multiple switches withsensors as exporting measuring devices;

FIG. 4 is a schematic of a feeder section with a breaker or switch withsensor as an importing measuring device;

FIG. 5 is a schematic of a line segment with series impedance and shuntadmittance;

FIG. 6 is a schematic of a phasor diagram with phase angles of residuavoltage differences at terminal buses of a line segment lagging thefaulty phase reference angle;

FIG. 7 is a schematic of a phasor diagram with phase angles of residualvoltage differences at terminal buses of a line segment leading thefaulty phase reference angle;

FIG. 8 is a schematic of a phasor diagram with phase angles of residualvoltage differences at terminal buses of a line segment leading thefaulty phase reference angle at one terminal bus, and lagging at otherterminal bus;

FIG. 9 is a schematic of a line segment with a single-phase-to-groundfault;

FIG. 10 is a schematic of a phasor diagram with phase angles of residualvoltage differences at a location along a line segment in phase with thefaulty phase reference angle; and

FIG. 11 is a block diagram of a fault locating method for a model of anungrounded distribution system according to some embodiments of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Ungrounded Distribution System and Fault Locating

FIG. 1 shows an example of ungrounded distribution system with asingle-phase-to-ground fault 104. The distribution system includes adistribution substation in which a three-phase transformer 102 receiveselectric power from power transmission systems, and provides the powerto downstream feeders via an upstream bus 101 and a downstream bus 103connected to the transformer 102.

The windings of the transformer 102 are ungrounded, either using WYE orDELTA connection. For example, in the FIG. 1, the primary winding of thetransformer uses the DELTA connection, and secondary winding uses WYEconnection. The feeder transfers powers to the loads through three-phasethree-wire lines. All loads can be DELTA connected. Each feeder can haveseveral switchable and measured devices, and measuring units attached tothose devices can provide three-phase voltage and three-phase currentmeasurements.

In the example of FIG. 1, the transformer 102 is connected to threefeeders, a feeder 110, a feeder 120 and a feeder 130. Each feeder caninclude one feeder breaker at its root, e.g., breakers 111, 121, and131. The feeders can also include switches defining sections of thefeeders. For example, the feeder 110 includes a switch 114 and a switch117. The feeder 120 includes a switch 124 and a switch 127. The feeder130 includes a switch 134 and a switch 137. The switchers can includesensors for measuring voltages, currents or both.

Distribution networks are typically of two types, radial orinterconnected. The distribution system 100 operates radially, i.e.,power leaves the station and passes through the network area with noconnection to any other power supply.

According to the location of switchable and measured devices, a feedercan be partitioned into several feeder sections. Each feeder section canhave one importing measuring device at the root of the section forproviding power to this section, and several exporting measuring devicesat the downstream boundaries of the section for providing power tosubsequent feeder sections. All line segments or devices between theimporting and the exporting measuring devices are part of the feedersection.

For example, the feeder 110 can be partitioned into three feedersections, section 112, section 115, and section 118. The feeder section112 includes all the line segments or devices between the upstream bus103 of breaker 111, and upstream bus 106 of switch 114, and includes oneimporting measuring device located at breaker 111, and one exportingmeasuring device located at switch 114. The section 115 is defined byall the line segments or devices between the upstream bus 106 of switch114, and upstream bus 108 of switch 117, and includes one importingmeasuring device located at switch 114, and one exporting measuringdevice located at switch 117. The feeder section 118 is defined as allline segments or devices downstream to the upstream bus 108 of switch117, and includes one importing measuring device located at switch 117.Feeder section 118 does not have any exporting measuring device.

The buses of a feeder sections can be partitioned into several layersaccording to the number of devices connected between each bus and theupstream bus of importing measuring device.

FIG. 3 and FIG. 4 show two examples of feeder sections with differentmeasuring conditions. FIG. 3 is a schematic of a feeder section 300 thathas multiple measuring devices at its boundaries. The feeder section 300has one importing measuring device at switch 310, and one exportingmeasuring device at switch 320. FIG. 4 is a schematic of a feedersection 400 that has only one measuring device, importing measuringdevice at switch 410.

The feeder section shown in FIG. 3 includes seven layers. Layers 1, 2,3, and 4 have one bus each. Layer 1 includes the bus 330 which is theupstream bus of importing measuring device. Layer 2, 3 and 4 includesthe buses 340, 350 and 360 respectively. Layer 5 includes the buses 370,374 and 378. Layer 6 and layer 7 have four buses each. Layer 6 includesthe buses 380, 382, 384, and 386. Layer 7 includes the buses 390, 392,394, and 396. The upstream bus of exporting measuring device, 384 isincluded in the layer 6.

The feeder section shown in the FIG. 4 can be partitioned into 6 layers.Layer 1 includes one bus 420, the upstream bus of importing measuringdevice 410. Layer 2 and 3 also have one bus each, bus 430, and 440respectively. Layer 4 has 3 buses, including bus 450, 454 and 458. Layer5 and layer six have four buses each. Layer 5 contains bus 460, 462,464, and 466. Layer 6 contains buses 470, 472, 474, and 476,

FIG. 2 shows a block diagram of a method 200 for locating asingle-phase-to-ground fault in an ungrounded distribution system. Stepsof the method can be implemented using a processor, connected to amemory and input output interfaces as known in the art.

The faulty phase 265 is determined 210 based on the phase-to-groundvoltage measurements 205. Then, the possible faulty area is narroweddown 220 and 230 to a specific feeder 275, and a specific feeder section285, based on the residual voltage measurements 215 and residual currentmeasurements 235. The residual voltages and residual currents may bemeasured directly or derived from phase-to-ground voltage measurements205, and phase current measurements 225.

After the faulty phase and faulty feeder section are known, the shuntcaused residual voltage distribution, and fault caused residual voltagedistribution Within the faulty feeder section are determined (240 and250) by using the phase-to-ground voltage measurements 205, residualcurrent measurements 235, system topology connectivity model 245, andseries impedance and shunt admittance models 255 of line segments.

Based those two types of residual voltage distribution within the faultyfeeder section, the fault location is further limited 260 to one orseveral specific line segments 280 based on the topology connectivitymodel 245, and the line impedance and admittance models 255. For eachpossible faulty line segment, a possible faulty location 290 isdetermined, 270 by testing a phase angle of residual voltage differenceat a location along the segment with the reference angle of the faultyphase. As commonly used in the art, and in this description, an angle ofthe voltage is a phase angle of the voltage.

A faulty line can be determined by comparing the angles of the voltageof the faulty phase with a faulty phase reference angle if assuming thatthe fault currents flow across the line segment instead of entering intothe ground through a location within the line segment. If the value ofthe reference angle of the faulty phase is between the angles of thevoltages on the bus terminals forming the line segment, i.e., anupstream bus and a downstream bus, then this line segment is faulty.However, to determine the voltage on the buses, the load of the busesshould be determined, which can be a difficult task. Thus, it is desiredto avoid the determination of the loads.

Some embodiments are based on a realization that a difference betweenthe fault caused residual voltage and the shunt caused residual voltageof the bus can approximate the voltage on the faulty phase of the bus.The fault caused residual voltage is a sum of the voltages of all threephases at the time of the fault determined under assumption that thecurrent flows across the line segment. This assumption leads to a changeof sign of the difference between the angle of the voltage on thedownstream bus and a reference angle of the faulty phase comparing to avoltage on the upstream bus of the faulty line.

A shunt caused residual voltage is a sum of the voltages of all threephases without fault determined based on shunt current, i.e., withoutthe fault current. Thus, the difference between the fault causedresidual voltage and the shunt caused residual can approximate thevoltage of the faulty phase. Because those voltages includes sum of thevoltages of the phases, and the loads are DELTA-connected, the loads ofdifferent phases cancel each other and thus do not have to bedetermined. Accordingly, determination of the faulty line is simplified.

The difference between the fault caused residual voltage and the shuntcaused residual voltage at the location of the fault is in phase with areference angle of the faulty phase. Thus, various location of thefaulty line segment can be tested with this equality to determine thelocation of the fault.

The measurements used in some embodiments of the invention are asteady-state power frequency components of voltage and currentmeasurements collected during the fault. Each measurement is describedby its magnitude, and phase angle. If the instantaneous waveforms areprovided instead of steady-state values at power frequency, then aleast-square regression method may be applied to extract the requiredpower frequency components from the instantaneous voltage and currentmeasurements. The measurements collected from the measuring units of afeeder breaker or switch with sensor include the currents flowingthrough the device downstream on phase a, b and c, I_(ps,a), I_(ps,b)and I_(ps,c), and the phase-to-ground voltages on phase a b and c,V_(p,a), V_(p,b) and V_(p,c), where bus p and s are the terminal busesof the breaker or switch, and bus p is upstream to bus s. Taken breaker111 in FIG. 1 as an example, the measurements include three-phasevoltages measured at its upstream bus 103, and three-phase currentsflowing through the breaker from its upstream bus 103 towards itsdownstream bus 105.

Based on the measured phase-to-ground voltages and phase currents, theresidual voltages and residual currents can be determined if notmeasured directly, according to:v _(p) ^(res) =V _(p,a) +V _(p,b) +V _(p,c),  (1)i _(ps) ^(res) =I _(ps,a) +I _(ps,b) +I _(ps,c),  (2)where, v_(p) ^(res) is the residual voltage at bus p, i_(ps) ^(res) isthe residual current flowing from bus p to bus s and measured at bus p.

Determination of Faulty Phase, Faulty Feeder and Faulty Section

The faulty phase is determined based on the measured phase-to-groundvoltages at the substation during the fault. During normal operation,the three phase-to-ground voltages are close to be balanced, that is,the normalized magnitudes of voltages are close to 1.0 per unit. When abolted single-phase-to-ground fault occurs at a feeder, thephase-to-ground voltage of the faulty phase of the faulty feeder, andadjacent feeders that connected to the same substation transformerinstantaneously approaches zero. On the other hand, due to theungrounded connection of the transformer, the substation maintains thephase-to-phase voltage close to unchanged, and then the phase-to-groundvoltages of the other two un-faulty phase instantaneously increases tovalues close to 1.73 times of its normal operation.

The phase-to-ground voltages measured at the secondary side of thesubstation transformer. i.e., the upstream bus of feeder breakers can beused to determine the faulty phase for a bolted single-phase-groundfault. The phase x is determined as faulty phase, if the followingconditions are met:|V _(sub,x) |≦Vxε{a,b,c}  (3)|V _(sub,y) ≧ Vyε{a,b,c},y≠x  (4)where, V_(sub,x) and V_(sub,y) are the phase-to-ground voltage measuredat the secondary side of the substation transformer on the phase x and yrespectively, and V and V are the lower and upper thresholds of voltagemagnitude used for abnormal voltage determination. For examples, V and Vcan be set as 0.30 per unit, and 1.40 per unit respectively.

The faulty feeder and faulty feeder section can be determined based onthe angle difference between residual voltage, and residual currents atthe measuring devices along the feeders. When the asymmetry ofdistribution power lines is ignored, the residual currents of anungrounded distribution system are contributed, by the phase-to-groundcapacities of un-faulty phases of the faulty and un-faulty feeders.

When a bolted single-phase-to-ground fault occurs at a location within afeeder, the residual voltage at the location increases to a value closeto three times of normal phase-to-ground voltage of the faulty phase.The direction of residual currents at a location downstream to thefaulty location, is flowing toward the substation, so the residualvoltage leads the residual current by around 90 degree. The direction ofresidual current at a location upstream to the fault is flowing towardthe faulty location and away from the substation, so the residualvoltage is lagging the residual current by around 90 degree.

Taken the single-phase-to-ground fault 104 in FIG. 1 as example, thefault 104 is within the feeder section 115 of feeder 110. Based on thenetwork topology, breaker 111 and switch 114 of feeder 110 are upstreamto the fault, and switch 117 is downstream to the fault. The residualcurrent of breaker 111 and switch 114 flows towards the faulty point, asshown by the directions of the hollow arrows 113 and 116. The residualcurrent of switch 117 flows towards the substation as shown by thedirection of hollow arrow 119. Similarly, the breakers and switches offeeder 120 and 130 are upstream to the faulty point, so the flowdirections of residual currents through those devices are towards thesubstation as shown by the hollow arrows 123, 126, 129, 133, 136 and139.

A feeder is determined as a faulty feeder, when the phase angledifference between the residual voltage and residual current measured atthe feeder breaker are close to 90 degree. Eq. (5) is used to determinewhether a feeder has a single-to-ground fault:|∠v _(fdr) ^(res) −∠i _(fdr) ^(res)−90°|<Δ θ,  (5)where, ∠v_(fdr) ^(res) is the phase angle of residual voltage measuredat the upstream terminal bus of the feeder breaker, ∠i_(fdr) ^(res) isthe phase angle of residual current flowing into the feeder breakerthrough its upstream terminal bus, Δ θ is a predetermined threshold ofangle difference according to the ratio of shunt sucesptance componentsover total shunt admittances of typical conductors used in thedistribution systems. For example, one embodiment sets Δ θ as 20 degree.

In the example of FIG. 1, if Eq. (5) is satisfied at the breaker 111,and not satisfied at breaker 121 and breaker 131, then the fault is atfeeder 110, and not at feeders 120 and 130.

The faulty section can be determined by checking the angle differencebetween the residual voltage and residual current of the measuringdevices at the boundaries of the feeder section. For example, a feedersection can be determined to be a faulty feeder section when the angledifference between residual voltage v_(im) ^(res) and residual currenti_(im) ^(res) at its importing measuring device are close to 90 degree,and the angle difference between residual voltage v_(ex) ^(res) andresidual current i_(ex) ^(res) at one of its exporting measuring deviceis close to −90 degree, that is, the following conditions are satisfied:|∠v _(im) ^(res) −∠i _(im) ^(res)−90°|<Δ θ,  (6)|∠v _(ex) ^(res) −∠i _(ex) ^(res)+90°|<Δ θ,  (7)

If the magnitude of residual current at the importing measuring deviceis close to zero, then only the exporting measuring devices can be usedto determine whether there is a fault within the section by using Eq.(7). For example, for a single-feeder substation, the residual currentmeasured at the feeder breaker is close to zero, so only themeasurements at the downstream switches are used. If a feeder sectionhas only one importing measuring device, whether it is a faulty sectionis determined by only using the measurements at the importing measuringdevice using Eq. (6).

In FIG. 1, if the angle differences are close to 90 degree at thebreaker 111 and switch 114, but −90 degree at the switch 117, then thefault is within the feeder section between switch 114 and switch 117,the feeder section 115 of feeder 110.

Determining Shunt Caused Residual Voltage Distribution

The shunt caused residual voltages are used to describe the residualvoltage distribution of a feeder section with given measured voltages atthe boundaries of the section if no fault occurs in the section.

The shunt caused residual voltage at bus p, {circumflex over (v)}_(p)^(res) can be determined based on the phase-to-ground voltages at bus p,according to:{circumflex over (v)} _(p) ^(res)=Σ_(xε{a,b,c}) {circumflex over (V)}_(p,x)  (8)where {circumflex over (V)}_(p,x) is the determined phase-to-groundvoltages of phase x at bus p.

For a feeder section, the determined voltage distribution within thesection can be determined based on the phase-to-ground voltagemeasurements at the importing and exporting measuring devices of thesection and its topology connectivity.

The voltages at buses connected to the measuring devices in the feedersection are directly set as the measured values:{circumflex over (V)} _(im) =V _(im),  (9){circumflex over (V)} _(ex) =V _(ex),  (10)where, {circumflex over (V)}_(im) and V_(im) are the determined andmeasured phase-to-ground voltages at the a boundary bus of an importingmeasuring device im, {circumflex over (V)}_(ex) and V_(ex) thedetermined and measured phase-to-ground voltages at a boundary bus ofthe exporting measuring device X.

The phase-to-ground voltages of a bus residing in the connectivity pathbetween each pair of the importing measuring device and one of theexporting measuring device are determined based on distances between thebus and two measuring devices and available voltage measurements at thetwo measuring devices. For example, the voltage of a bus residing on apath between the boundary buses of the two measuring devices can bedetermined as a weighted average of the voltages of the boundary busesweighted proportionally to relative distances between the bus on thepath and the boundary buses.

For example, the phase-to-ground voltage of bus p can be determinedaccording to:

$\begin{matrix}{{\hat{V}}_{p} = {{\frac{d_{p - {ex}}}{d_{{im} - p} + d_{p - {ex}}}V_{im}} + {\frac{d_{{im} - p}}{d_{{im} - p} + d_{p - {ex}}}V_{ex}}}} & (11)\end{matrix}$where, {circumflex over (V)}_(p) is the vector of determinedphase-to-ground voltages of bus p. V_(im) and V_(ex) are the vectors ofphase-to-round voltages measured at the importing and exportingmeasuring devices, d_(im-p) and d_(p-ex) are the sum of length of linesegments at the path between the bus and the upstream buses of importingdevice, and exporting device respectively.

If there are multiple exporting measuring devices for the feedersection, and common buses between different paths, then the voltages ofthose common buses can be set as the average of determined voltages forall paths:

$\begin{matrix}{{{\hat{V}}_{p} = {\frac{1}{m}{\sum\limits_{i = 1}^{m}\;\left( {{\frac{d_{p - {ex}_{i}}}{d_{{im} - p} + d_{p - {ex}_{i}}}V_{im}} + {\frac{d_{{im} - p}}{d_{{im} - p} + d_{p - {ex}_{i}}}V_{{ex}_{i}}}} \right)}}},} & (12)\end{matrix}$where, in is the total number of paths that pass through bus p, V_(ex)_(i) is the measured voltage of i-th exporting measuring device ex_(i),and d_(p-ex) _(i) is the sum of length of line segments at the pathbetween the bus p and the upstream bus s of the i-th exporting measuringdevice.

For any bus not on any paths between the measuring devices, but fed fromone of buses in the paths, its voltage can be determined as the voltageof the feeding bus on the paths according to:{circumflex over (V)} _(s) ={circumflex over (V)} _(p),  (13)where, bus s is a bus not in the paths, bus p is a bus in the paths,{circumflex over (V)}_(s) is vector of the determined phase-to-groundvoltages of bus s.

FIG. 3 shows an example of the feeder section having one importingmeasuring device at bus 330, and one exporting measuring device at 384.All buses on the path between the importing and exporting measuringdevices, such as bus 340, 350, 360 and 374 are determined based on itsdistance to bus 330 and 384 and measured voltages at bus 330 and 384according to Eq. (11). The voltages of all buses downstream bus 360,including, bus 370, 382, 390, 392 can be determined, the same as thevoltage of bus 360. The voltages of all buses downstream bus 374,including bus 378, 386, 388, 394, and 396 can be determined the same asthe determined voltage of bus 374.

For a feeder section with only one importing measuring device, all buseswithin the section can be set as the measured voltages at the importingmeasuring device. For example, in FIG. 4, the feeder section has onlyone importing measuring device at bus 420, the voltages of all buses inthe section are set as the same as the voltages measured at bus 420.

Determining Shunt Caused Residual Current Distribution

If there is no fault in the feeder section, then the residual currentson each line segment in the feeder section are solely contributed by theshunt currents of line segments downstream to the importing measuringdevice of the feeder section. The shunt currents can be determined basedon the determined voltages at the terminal buses of the associated linesegments and shunt admittance of those segments. This type of residualcurrents is called shunt caused residual currents. Some embodimentsignore the asymmetry of the distribution lines in determining the shuntcaused residual currents.

In some embodiments of the invention, the contributions of residualcurrents from each phase are modeled and solved independently. For aline segment between bus p and bus s, the residual current from bus p tobus s, i_(ps) ^(res) can be determined according toi _(ps) ^(res)=Σ_(xε{a,b,c}) I _(ps,x) ^(res)  (14)where, I_(ps,x) ^(res) is the residual current component of line segmentbetween bus p and bus s contributed from phase x. And the residualcurrents contributed from each phase can also be represented as avector, I_(ps) ^(res):

$\begin{matrix}{I_{p\; s}^{res} = {\begin{bmatrix}I_{{p\; s},a}^{res} \\I_{{p\; s},b}^{res} \\I_{{p\; s},c}^{res}\end{bmatrix}.}} & (15)\end{matrix}$

A backward sweep method can be used to determine the shunt causedresidual current distribution in the feeder section. The method startsat the line segments connected with the last layer of the feedersection, then moves backward to next layer upstream, and ends at theline segments connected to the first layer of the section. For each linesegment, the residual currents of a line segment at the downstream busside is determined first based on the residual currents of downstreamline segments, then the residual currents at the upstream bus side isdetermined by the values at the downstream side and the shunt currentscontributed from the shunt admittances of the line segment.

For the example shown in FIG. 3, the backward sweep method starts fromthe line segments connected upstream to the last layer, layer 7 firstly,that is line segment between 380 and 390, 380 and 392, 386 and 394, andbetween 386 and 396. Then moving to line segments connected upstream tolayer 6, including line segments between 370 and 380, 370 and 382, 374and 384, 378 and 386, and between 378 and 388. The process is endingwhen the currents of switch between 330 and 340 connected upstream tolayer 2 are determined.

FIG. 5 shows an example of a line segment 500 between a upstream bus p510 and a downstream bus s, 520. The line segment is modeled by a seriesphase impedance matrix Z_(ps) ^(se) 530, and a shunt admittance matrixY_(ps) ^(sh) partitioned into two terminal buses, 540, and 550. Thephase-to-ground voltages at bus/2 and bus s are represented by thevectors V_(p) 516 and V_(s) 526, and the residual voltages at bus p andbus s are represented by the variables v_(p) ^(res) 512 and v_(s) ^(res)522. The phase residual currents flowing on the line segments arerepresented by the vectors I_(ps) ^(res) 518 and I_(ps′) ^(res) 528,I_(ps) ^(res) 518 is the vector of residual currents entering into theline segment through bus p 510, and I_(ps′) ^(res) 528 is the vector ofresidual currents leaving from the line segment through bus s 520.

For a line segment connected an upstream bus p to a downstream bus s,the shunt caused residual current leaving the segment through thedownstream bus s, Î_(ps′) ^(res) is determined as:Î _(ps′) ^(res)=Σ_(tεDD) _(sÎ) _(st) ^(res)  (16)where, DD_(s) is the set of downstream buses that directly connectedwith bus s through a line segment or switch, and Î_(st) ^(res) is thevector of shunt caused residual currents flowing into a line segmentbetween bus s and bus t through bus s. The shunt caused residualcurrents entering the segment between bus p and bus s through theupstream bus p is determined as:Î _(ps) ^(res) =Î _(ps′) ^(res)+½Y _(ps) ^(sh)({circumflex over (V)}_(p) +{circumflex over (V)} _(s))  (17)where, {circumflex over (V)}_(p) and {circumflex over (V)}_(s) are thevectors of determined voltages of bus p and bus s.

For a switch connected an upstream bus p to a downstream bus s, Eq. (16)is used to determine the shunt caused residual currents at thedownstream bus side, and the shunt caused residual currents at upstreambus side are set the same as the downstream one:Î _(ps) ^(res) =Î _(ps′) ^(res)  (18)

If the downstream bus s of a line segment between bus p and s is anupstream bus of an exporting measuring device, then the shunt causedresidual currents at the downstream bus side are determined based on themeasurements at the exporting measuring device, and shunt currentsdownstream to the exporting device according to:Î _(ps′,x) ^(res) =i _(ex) ^(res)−Σ_(y≠x,yε{a,b,c}) Î _(ps′,y)^(res),  (19)Î _(ps′,y) ^(res)=Σ_(mnεDN) _(ex) ½Y _(mn,y) ^(sh)({circumflex over (V)}_(m) +{circumflex over (V)} _(n)),  (20)where, Î_(ps′,x) ^(res) and Î_(ps′,y) ^(res) are the shunt causedresidual currents leaving the line segment between bus p and bus sthrough bus s on the faulty phase x, and one of un-faulty phase v,i_(ex) ^(res) is the residual current measurement at the exportingmeasuring device, DN_(X) is the set of all line segments downstream tothe exporting measuring device, Y_(mn,y) ^(sh) is the vector of shuntadmittance elements of line segment between bus in and bus n at the rowcorresponding to the un-faulty phase v, {circumflex over (V)}_(m) and{circumflex over (V)}_(n) are the determined voltages of bus in and busn.

Determining of Fault Caused Residual Current Distribution

If a fault is occurring downstream to a line segment, then the residualcurrent of the faulty phase of the line segment is mainly contributedfrom the shunt currents on the un-faulty phases of the faulty feeder andadjacent feeders connected to the same substation transformer with thefaulty feeder. This type of residual currents is called fault causedresidual currents.

The fault caused residual currents can be determined for each linesegment in the faulty section, and can be used to determine acorresponding fault caused voltage for determining the possible faultyline segments.

The distribution of fault caused residual currents are determined basedon the residual current measurements at the root of the faulty section,the determined phase-to-ground voltages and shunt admittances of linesegments through a forward sweep method. The faulted caused residualcurrents of line segments or devices are calculated by sequentiallyassuming a fault at the downstream terminal bus of a line segment ordevice, starting from the devices connected to the root of the feedersection and towards the ends of the feeder section.

Using FIG. 4 as an example, the method is started from the deviceconnected downstream to bus 420 in the layer 1, switch 410 connected bus420 to bus 430. Then moving on to the line segment connected to the busof layer 2, i.e., the line segment between bus 430 and bus 440. Thismethod ends when the residual current calculation for the line segmentsbetween layer 5 and layer 6 are completed.

For any line segment connecting an upstream bus p to a downstream bus s,the fault caused residual current entering the segment through theupstream bus p, Ĩ_(ps) ^(res) is determined according to:Ĩ _(ps) ^(res) =Ĩ _(dp′) ^(res)−Σ_(tεDD) _(p) _(,t≠s) Ĩ _(pt)^(res),  (21)where, Ĩ_(dp′) ^(res) is the fault caused residual current leaving anupstream line segment connected bus d to bus p through bus p, DD_(p) isthe set of buses downstream to bus p and having direct connection withbus p through a line segment or a switch, Ĩ_(pt) ^(res) the shunt causedresidual current entering the line segment or switch between bus p and tthrough bus p.

The fault caused residual current leaving the line segment through buss, Ĩ_(ps′) ^(res) can be determined as:Ĩ _(ps′) ^(res) =Ĩ _(ps) ^(res)−½ Y _(ps) ^(sh)({circumflex over (V)}_(p) +{circumflex over (V)} _(s)),  (22)where, {circumflex over (V)}_(p) and {circumflex over (V)}_(s) are thedetermined voltages of bus p and bus s.

For a switch between bus p and bus s, Eq. (21) can be used to determinethe fault caused residual currents at the upstream side, and then usingthose values to set the ones at downstream bus directly:Î _(ps′) ^(res) =Ĩ _(ps) ^(res).  (23)

If the upstream bus of a line segment or switch between bus p and s isan upstream bus of an importing measuring device, the residual currentsat the upstream bus side are determined based on the measurements at theimporting device, and shunt currents downstream to the importing deviceaccording to:Î _(ps′,x) ^(res) =i _(im) ^(res)−Σ_(y≠x,yε{a,b,c}) Î _(ps,y)^(res),  (24)Î _(ps,y) ^(res)=Σ_(mnεDN) _(im) ½Y _(mn,y) ^(sh)({circumflex over (V)}_(m) +{circumflex over (V)} _(n)),  (25)where, Ĩ_(ps,x) ^(res) and Ĩ_(ps,y) ^(res) are the fault-caused residualcurrents entering the line segment between bus p and bus s through bus pon the faulty phase x, and one of un-faulty phase y, i_(im) ^(res) isthe residual current measurements at the importing measuring device ini,DN_(im) is the set of all line segments downstream to the importingmeasuring device im.

Determination of Faulty Line Segments

In some embodiments of the invention, the faulty line segment between anupstream and a downstream terminal buses is determined by comparing afirst angle of an angle difference between angle of the fault causedresidual voltage and the angle of the shunt caused residual voltage onthe upstream bus and a second angle of the difference between an angleof the fault caused residual voltage and an angle of the shunt causedresidual voltage on a downstream bus. For example, some embodimentscompare the first and the second angles with a reference angle of thefaulty phase. In these embodiments, the faulty segment is determinedwhen the value of the reference angle of the faulty phase is between thevalues of the first and the second angles.

The fault caused residual voltages of terminal buses are determinedbased on the residual voltages and residual currents measured at theroot of the faulty section, fault caused residual currents, and lineseries impedances and shunt admittances of line segments.

A forward sweep method can be used to determine the fault causedresidual voltages for each line segment starting from the root of thefeeder section towards to the end of the feeder section.

For any line segment connected an upstream bus p to a downstream bus s,the fault caused residual voltage at the downstream bus is determinedbased on the one at the upstream bus, and the fault caused currententering the line segment through the upstream bus according to:{tilde over (v)} _(s) ^(res) ={tilde over (v)} _(p) ^(res) −T ^(T) Z_(ps) ^(se)(Ĩ _(ps) ^(res)−½Y _(ps) ^(sh) {circumflex over (V)}_(p))  (26)where, {tilde over (v)}_(s) ^(res) and {tilde over (v)}_(p) ^(res) arethe fault caused residual voltage at bus s and bus p, T is a vector ofunity defined as T=[1 1 1]^(T), Ĩ_(ps) ^(res) is the vector of faultcaused residual currents entering, the line segment though bus p,{circumflex over (V)}_(p) is the vector of determined phase-to-groundvoltages at bus p.

For any switch between bus p and bus s, the fault caused residua voltageat the downstream bus is set as the same as of the upstream bus:{tilde over (v)} _(s) ^(res) ={tilde over (v)} _(p) ^(res)  (27)

The forward sweep method is starting from the upstream bus of importingmeasuring device at the root of the feeder section. The measuredresidual voltage is used to set the fault caused voltage at the bus ofimporting device:{tilde over (v)} _(im) ^(res) =v _(im) ^(res)  (28)where, {tilde over (v)}_(im) ^(res) is the fault caused residual voltageat upstream bus of importing device im, v_(im) ^(rea) is the residualvoltage measured at importing device im during the fault.

The faulty line segment is identified by comparing the angles of thedifference of fault caused residual voltage and shunt caused residualvoltage for two terminal buses of the segment. If the angles at twoterminal buses are located at different sides of a reference axisdefined by a reference angle according to the faulty phase, then theline segment is determined to be a faulty segment.

In some embodiments, Eq (29) is used to determine whether a line segmenthas a fault:sin(∠({tilde over (v)} _(p) ^(rea) −{tilde over (v)} _(p) ^(res))−θ_(x)^(ref))sin(∠({tilde over (v)} _(s) ^(res) −{circumflex over (v)} _(s)^(res))−θ_(x) ^(ref))<0,  (29)where, ∠({tilde over (v)}_(p) ^(res)−{circumflex over (v)}_(p) ^(res))and ∠({tilde over (v)}_(s) ^(res)−{circumflex over (v)}_(s) ^(res)) arethe phase angle of the difference of fault caused and shunt causedresidual voltages at bus p, and bus s respectively, and θ_(x) ^(ref) isthe reference angle determined by the faulty phase.

In some embodiments, the reference angle of the faulty phase is set asthe phase angle of internal voltage of equivalent infinite source fortransmission systems fed to the substation. For example, the referenceangle can be set as 0, −120 or 120 degree if the faulty phase is phasea, phase b, or phase c respectively.

FIGS. 6, 7 and 8 show three examples of phasor diagrams for residualvoltage differences at terminal buses of a line segment.

In FIG. 6, both the angles of the fault-caused and shunt-caused residualvoltage differences at terminal bus p and bus s, i.e., the first angle610 and the second angle 620 are lagging the reference angle of thefaulty phase θ_(x) ^(ref) 630. Thus, for the line segments shown in FIG.6, there is no fault occurring within the segments.

Similarly as; shown in FIG. 7, the angles of the fault-caused andshunt-caused residual voltage differences at terminal bus p and busi.e., the first angle 710 and the second angle 720 are leading thereference angle of the faulty phase θ_(x) ^(ref) 730. Thus, for the linesegments shown FIG. 7, there is no fault occurring within the segments.

In FIG. 8, the angle of the fault-caused and shunt-caused residualvoltage difference at bus p, 810 is lagging the reference angle of thefaulty phase θ_(x) ^(ref) 830, and the angle of the fault-caused andshunt-caused residual voltage difference at bus s, 820 is leading thereference angle of the faulty phase θ_(x) ^(ref), 830. So, there ispossibly a fault occurring at the line segment.

Determination of Fault Location Based on Residual Voltages

After the faulty line segments are determined, the possible faultylocations along each line segment can be determined according thedifference of the angle of fault caused residual voltage and the angleof shunt caused residual voltage.

Specifically, the difference between the fault caused residual voltageand the shunt caused residual at the location of the fault is in phasewith a reference angle of the faulty phase. Thus, various location, ofthe faulty line segment can be tested with this equality to determinethe location of the fault.

FIG. 9 shows a model of a line segment with a phase-to-ground fault. Theline, segment 900 connects an upstream bus p 910 to a downstream bus s915, and is partitioned into two subsegments according to the locationof the fault f 932. The first segment is between bus p 910 and thelocation of fault f′932, and the second segment is between location ofthe fault f 932 and bus s 915.

For a ratio d of the distance between the fault location f 932 and theupstream bus p 910 over total length of the line segment between bus p910 and bus s 915, the first segment can be modeled with a seriesimpedance dZ_(ps) ^(se), 940 and a shunt admittance dY_(ps) ^(sh)partitioned into two terminal buses, 950 and 960, and the second segmentcan be modeled with a series impedance (1−d)Z_(ps) ^(se), 970 and ashunt admittance (1−d)Y_(ps) ^(sh) partitioned into terminal buses 980and 990.

Based on the series impedance and shunt impedance model for the firstsegment between bus p and fault point f, the fault caused residualvoltage can be calculated as:

$\begin{matrix}{{{\overset{\sim}{v}}_{f}^{res} = {{\overset{\sim}{v}}_{p}^{res} - {{dT}^{T}{Z_{p\; s}^{se}\left( {{\overset{\sim}{I}}_{p\; s}^{res} - {\frac{d}{2}Y_{p\; s}^{sh}{\hat{V}}_{p}}} \right)}}}},} & (30)\end{matrix}$where {tilde over (v)}_(f) ^(res) is the fault caused residual voltageat the location f, {tilde over (v)}_(p) ^(res) and Ĩ_(ps) ^(rea) are thefault caused residual voltage at upstream bus p, and the vector of faultcaused residual currents entering the line segment through bus p, and{circumflex over (V)}_(p) is the vector of determined phase-to-groundvoltages of bus p.

Similarly based on the series impedance and shunt impedance model forthe second segment between fault location f and bus s, the shunt causedresidual voltage can be calculated as:

$\begin{matrix}{{{\hat{v}}_{f}^{res} = {{\hat{v}}_{s}^{res} + {\left( {1 - d} \right)T^{T}{Z_{p\; s}^{se}\left( {{\hat{I}}_{p\; s^{\prime}}^{res} + {\frac{1 - d}{2}Y_{p\; s}^{sh}{\hat{V}}_{s}}} \right)}}}},} & (31)\end{matrix}$where, {circumflex over (v)}_(f) ^(res) and {circumflex over (v)}_(s)^(rea) are the shunt caused residual voltages at fault location f anddownstream bus s, Î_(ps′) ^(res) is the vector of shunt-caused residualcurrents leaving the line segment through bus s, {circumflex over(V)}_(s) is the vector of determined phase-to-ground voltages of bus s.

Some embodiments compare the angle of the difference of fault causedresidual voltages and shunt caused residual voltages with the referenceangle corresponding to the fault phase. If those angles are closeenough, then the fault is located at the location f.

For example, some embodiments use Eq. (32) to determine whether alocation f is faulty:|({tilde over (v)} _(f) ^(res) −{circumflex over (v)} _(f) ^(res))−θ_(x)^(ref)|<ε,  (32)where, ε is a small threshold, such as 0.00001.

FIG. 10 shows an example of phasor diagram for a location f within aline segment between the bus p and the bus s that having a phase angleof residual voltage differences in phase with the faulty phase basedreference angle. This location f can be determined possibly having abolted single-phase-to-ground fault at the faulty phase.

As shown in FIG. 10, the difference of fault-caused and shunt-causedresidual voltages at bus p, 1010 lags the reference angle axis 1040, thedifference of residual voltages at bus s, 1020 leads the reference angleaxis 1040, and the difference of residual voltages at the location f,1030 is in phase with the reference angle axis 1040.

EXAMPLE

FIG. 11 shows a block diagram of a method for locating a boltedsingle-to-ground fault of an ungrounded distribution system according toone embodiment of the invention. Various embodiments of invention use atleast part of the steps of the method 1100.

In step 1105, the phase-to-ground voltage and phase current measurementsat the feeder breakers of feeders in the substation are determined,e.g., retrieved from the measurement units.

In step 1110, the residual voltage and residual current of the feederbreaker for each feeder in the substation are determined.

In step 1115, the faulty phase is determined by comparing the magnitudesof phase-to-ground voltage measurements of a feeder breaker with theupper and the lower thresholds.

In step 1120, the faulty feeder is determined by comparing the phaseangle difference between residual voltage and residual current of thefeeder breaker for each feeder in the substation.

In step 1125, the phase-to-ground voltages and phase currentmeasurements are determined, e.g., received, for the switches withsensors along the faulty feeder.

In step 1130, the residual voltages and residual currents for theswitches with sensors along the faulty feeder are determined.

In step 1135, the faulty feeder section of the faulty feeder isdetermined, by comparing the phase angle difference between residualvoltages and residual currents for the feeder breaker, and the switcheswith sensors along the faulty feeder.

In step 1140, the phase-to-ground voltage distribution of the faultyfeeder section is determined based on the phase-to-ground voltagemeasurements at the feeder breaker or switches and feeder topologyconnectivity.

In step 1145, the shunt caused residual voltage distribution of thefaulty feeder section is determined based on the determinedphase-to-ground voltages.

In step 1150, determine the shunt caused residual current distributionof line segments within the faulty feeder section using a backward sweepmethod. The method starts from the line segments connected to the endsof the section towards the root of the section based on determinedphase-to-ground voltages and shunt admittances of line segments.

In step 1155, determine the fault caused residual current distributionof line segments in the faulty feeder section using a forward sweepmethod. The sweep method starts from the line segments connected to theroot of the section towards the ends of the section based on theresidual current measurements at the root of the section, shunt causedresidual currents determined in step 1150, and shunt admittances of linesegments.

In step 1160, determine the fault caused residual voltage distributionof line segments in the faulty feeder section, starting from the root ofthe faulty feeder section towards the ends of the section based on theresidual voltages measured at the root of the section, fault Causedresidual currents determined in step 1155, shunt caused residualcurrents in step 1150, and series impedance and shunt admittance of linesegments.

In step 1165, determine the possible faulty line segments by comparingthe angle changes for the difference of the fault caused residualvoltage, and the shunt caused residual voltage between two terminalbuses of each line segment.

In step 1170, determine the possible fault location for each possiblefaulty line segment by finding a location along the line segment thathas a angle of the difference between the fault caused residual voltageand the shunt caused residual voltage equals to a reference anglecorresponding to the faulty phase.

In step 1175, output the fault locating results, including faulty phase,faulty feeder, faulty feeder section, possible faulty line segments, andpossible geographic locations of faulty locations to the distributionautomation systems.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

The invention claimed is:
 1. A method for determining a location of afault in an ungrounded power distribution system, wherein the powerdistribution system includes a set of feeders connected to a substation,wherein each feeder includes a set of loads connected by line segmentsand each line segment includes an upstream bus and a downstream bus, andthe fault is a bolted single-phase-to-ground fault, comprising:determining shunt caused residual voltages on the upstream bus and thedownstream bus of the line segment within a faulty feeder section of afaulty feeder; determining fault caused residual voltages on theupstream bus and the downstream bus of the line segment; designating theline segment as a faulty line segment when a reference angle of a faultyphase is between a first angle of a difference between an angle of thefault caused residual voltage and an angle of the shunt caused residualvoltage on the upstream bus and a second angle of a difference betweenan angle of the fault caused residual voltage and an angle of the shuntcaused residual voltage on the downstream bus; and determining alocation of a point on the faulty line segment with a difference betweenthe angle of the fault caused residual voltage and the angle of theshunt caused residual voltage in phase with the reference angle of thefaulty phase as the location of the fault, wherein steps of the methodare performed by a processor.
 2. The method of claim 1, furthercomprising: determining a shunt caused residual voltage distribution anda fault caused residual voltage distribution on the faulty feedersection using voltages and currents measured for feeder breakers atroots of the feeders, and voltages and currents measured for switcheswith sensors along the feeders during the fault; determining the shuntcaused residual voltages on the upstream bus and the downstream bus ofthe line segment using the shunt caused residual voltage distribution;and determining the fault caused residual voltages on the upstream busand the downstream bus of the line segment using the fault causedresidual voltage distribution.
 3. The method of claim 1, furthercomprising: determining the reference angle of the faulty phase as aninternal phase angle of an equivalent infinite source fed into asubstation of the ungrounded power distribution system at the faultyphase.
 4. The method of claim 1, further comprising: comparingphase-to-ground voltages measured at roots of each feeder in the set offeeders with lower and upper thresholds to determine the faulty phase;comparing a difference between an angle of a residual voltage and anangle of a residual current measured at a root of each feeder from theset of feeders with a angle threshold to determine the faulty feeder;and comparing a difference between an angle of a residual voltage and anangle of a residual current measured at boundaries of each feedersection of the faulty feeder with the angle threshold to determine thefaulty feeder section.
 5. The method of claim 1, further comprising:determining the shunt caused residual voltage for each bus in the faultyfeeder section as a sum of phase-to-ground voltages for all phases atthe bus.
 6. The method of claim 5, wherein the phase-to-ground voltagesfor each bus in the faulty feeder section are determined based onvoltages measured at the boundaries of faulty feeder section and lengthsof each line segment in the faulty feeder section, further comprising:determining voltages of boundary buses at the boundaries of the faultyfeeder section using measurements provided by an importing measuringdevice and an exporting measuring device; determining a voltage of a buson a path between the boundary buses as a weighted average of thevoltages of the boundary buses weighted proportionally to relativedistances between the bus on the path and the boundary buses; anddetermining a voltage of a bus not on the path as a voltage of a feedingbus on the path.
 7. The method of claim 1, further comprising:determining shunt caused residual currents on each phase of linesegments of the faulty feeder section by accumulating shunt currentsgenerated from shunt admittances of the line segments, wherein theaccumulating is performed sequentially for each line segment in anupstream direction starting from a line segment connected to the end ofthe faulty feeder section and moving towards the root of the faultyfeeder section.
 8. The method of claim 7, further comprising:initializing the shunt caused residual currents on each phase at theexporting measuring devices of the faulty section with the residualcurrents measured at the exporting devices, and shunt currents on theun-faulty phases generated by shunt admittances of all line segmentsdownstream to the exporting devices and determined based on thephase-to-ground voltages of buses of the line segments.
 9. The method ofclaim 1, further comprising: determining fault caused residual currentsfor line segments in the faulty feeder section by sequentially assuminga fault downstream to the line segment starting from a line segmentconnected to a root of the faulty feeder section and moving towards theend of the feeder section.
 10. The method of claim 9, furthercomprising: initializing the faulted caused residual currents on eachphase of a device connected to a importing measuring device of thefaulty feeder section with the residual current measured for theimporting device, and shunt currents on the un-faulty phases generatedby shunt admittances of all line segments downstream to the importingdevices.
 11. The method of claim 9, further comprising: determiningfault caused residual currents for the faulty phase and un-faulty phasesof a line segment separately based on a difference of a fault causedresidual current on an upstream line segment and the shunt causedresidual currents on adjacent line segments and the shunt currentsgenerated by the shunt admittance of the line segment.
 12. The method ofclaim 1, further comprising: determining the fault caused residualvoltages of the the upstream bus and the downstream bus of the linesegment in the faulty feeder section by sequentially assuming a faultdownstream to the line segment starting from a line segment connected toa root of the faulty feeder section and moving towards the end of thefeeder section.
 13. The method of claim 12, further comprising:determining the fault caused residual voltages of the downstream bus ofa line segment based on the fault caused residual voltage at theupstream bus of the line segment and the fault caused residual currentsfor each phase by using a series impedance and a shunt admittance of theline segment.
 14. The method of claim 1, further comprising: determiningthe faulty line segment when the reference angle is between the firstangle and the second angle according tosin(∠({tilde over (v)} _(p) ^(res) −{circumflex over (v)} _(p)^(res))−θ_(x) ^(ref))sin(∠({tilde over (v)} _(s) ^(res) −{circumflexover (v)} _(s) ^(res))−θ_(x) ^(ref))<0, where, ∠({tilde over (v)}_(p)^(res)−{circumflex over (v)}_(p) ^(res)), is the first angle, ∠({tildeover (v)}_(s) ^(res)−{circumflex over (v)}_(s) ^(res)) is the secondangle, θ_(x) ^(ref) is the reference angle determined by the faultyphase, {tilde over (v)}_(s) ^(res) and {tilde over (v)}_(p) ^(res) arethe fault caused residual voltage at the downstream bus s and theupstream bus p, and {circumflex over (v)}_(s) ^(res) and {circumflexover (v)}_(p) ^(res) are the shunt caused residual voltage at thedownstream bus s and the upstream bus p.
 15. The method of claim 1,further comprising: determining the location of the fault by testinglocations along the faulty line segment according to|({tilde over (v)} _(f) ^(res) −{circumflex over (v)} _(f) ^(res))−θ_(x)^(ref)|<ε, wherein ε is a threshold, {circumflex over (v)}_(f) ^(res) isa shunt caused residual voltage at the location f of the fault, {tildeover (v)}_(f) ^(res) is a fault caused residual voltage at the locationf of the fault.
 16. The method of claim 1, further comprising:determining the fault caused residual voltage at a location of a pointon the faulty line segment based on the fault caused residual voltageand current at the upstream bus of the line segment, and a seriesimpedance and a shunt admittance of a portion of line segment from theupstream bus of the line segment to the location of the point.
 17. Themethod of claim 1, further comprising: determining the shunt causedresidual voltages at a location of a point on the faulty line segmentbased on the residual voltage and the residual current for each phase atthe downstream terminal bus of the line segment, and a series impedanceand a shunt admittance of a portion of line segment from the location ofthe point to the downstream terminal bus.
 18. A system for determining alocation of a fault in an ungrounded power distribution system, whereinthe power distribution system includes a set of feeders connected to asubstation, wherein each feeder includes a set of loads connected byline segments and each line segment includes an upstream bus and adownstream bus, and the fault is a bolted single-phase-to-ground fault,comprising a processor for: determining shunt caused residual voltageson the upstream bus and the downstream bus of the line segment within afaulty feeder section of a faulty feeder; determining fault causedresidual voltages on the upstream bus and the downstream bus of the linesegment; determining a faulty line segment as a line segment with avalue of a reference angle of a faulty phase between a first angle and asecond angle, wherein the first angle equals a difference between anangle of the fault caused residual voltage and an angle of the shuntcaused residual voltage on the upstream bus, and wherein the secondangle equals a difference between an angle of the fault caused residualvoltage and an angle of the shunt caused residual voltage on thedownstream bus; and determining a location of a point on the faulty linesegment with a difference between the angle of the fault caused residualvoltage and the angle of the shunt caused residual voltage in phase withthe reference angle of the faulty phase as the location of the fault.